Reducibility of Quasi-periodic Linear KdV Equation
Reducibility of Quasi-periodic Linear KdV Equation
复制标题
准周期线性KdV方程的约简性
DOI：
10.1007/s10884-020-09916-6
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发表时间：
2020
影响因子：
1.3
通讯作者：
Yi Yingfei
中科院分区：
文献类型：
--
作者：
Geng Jiansheng;Ren Xiufang;Yi Yingfei
在本文中，我们考虑下列一维拟周期强迫线性KdV方程$$\开始{aligned} u_t+（1+ a_{1}（\omega t））u_{xxx}+ a_{2}（\omega t，x）u_{xx}+ a_{3}（\omega t，x）u_{x} +a_{4}（\omega t，x）u=0 \end{aligned}$$在周期边界条件下，其中s是位于有界闭区域内的频率向量对于某些，，，，，是真实的解析的，在一个合适的范数下，由一个小参数从上面界定，并且是偶数，是奇数。在系数的真实的解析性假设下，我们重新访问Baldi等人的结果（Math Ann 359（1-2）：471-536，2014），通过显示存在Cantor seý，使得对于每个，对应的方程可以平滑地约简为常数系数方程。我们的主要结果去除了Baldi et al.（2014）中最初假设的条件，因此可以得到可逆的，拟周期强迫的，非线性KdV方程的拟周期解的普遍存在性和线性稳定性结果，对非线性的限制要少得多。我们的约化结果的证明利用了方程的一些特殊结构，并基于对变系数同调方程解的精化Kuksin估计。
In this paper, we consider the following one-dimensional, quasi-periodically forced, linear KdV equations $$\begin{aligned} u_t+(1+ a_{1}(\omega t)) u_{xxx}+ a_{2}(\omega t,x) u_{xx}+ a_{3}(\omega t,x)u_{x} +a_{4}(\omega t,x)u=0 \end{aligned}$$under the periodic boundary condition, where’s are frequency vectors lying in a bounded closed regionfor some,,,, are real analytic, bounded from the above by a small parameterunder a suitable norm, andare even,are odd. Under the real analyticity assumption of the coefficients, we re-visit a result of Baldi et al. (Math Ann 359(1–2):471–536, 2014) by showing that there exists a Cantor setwithsuch that for each, the corresponding equation is smoothly reducible to a constant-coefficient one. Our main result removes a condition originally assumed in Baldi et al. (2014) and thus can yield general existence and linear stability results for quasi-periodic solutions of a reversible, quasi-periodically forced, nonlinear KdV equation with much less restrictions on the nonlinearity. The proof of our reducibility result makes use of some special structures of the equations and is based on a refined Kuksin’s estimate for solutions of homological equations with variable coefficients.